Download 5.3 Parallel Lines and Congruent Angles

3.1 PARALLEL LINES
Thompson
How can I recognize planes, transversals, pairs of
angles formed by a transversal, and parallel lines?
What You Should Learn
Why You Should Learn It
Goal 1: How to identify angles formed by two
lines and a transversal
 Goal 2: How to use properties of parallel lines

Angles Formed by a Transversal
Transversal – a line that intersects two or more
coplanar lines at different points
 In the figure, the transversal t intersects the
lines L and M

t
L
M
Corresponding Angles

Two angles are corresponding angles if they occupy corresponding
positions, such as 1 and 5
t
4
1
3 2
L
M
8 5
7 6
Alternate Interior Angles

Two angles are alternate interior angles if they lie between L and M
on opposite sides of t, such as 2 and 8
t
4
1
3 2
L
M
8 5
7 6
Alternate Exterior Angles

Two angles are alternate exterior angles if they lie outside L and M
on opposite sides of t, such as 1 and 7
t
4
1
3 2
L
M
8 5
7 6
Same-Side Interior Angles (consecutive)

Two angles are same-side interior angles if they lie between L and
M on the same side of t, such as 2 and 5
t
4
1
3 2
L
M
8 5
7 6
These definitions depend on what line is
the transversal.


Before you define the angle terms, you need to think
about which line is the transversal!
Even within the same diagram, the transversal can
change. This will often lead to “ignoring” other parts of
the diagram (as we’ll see next…)
Example 1
Naming Pairs of Angles

How is
9
related to the other angles?
n
4
1
m
3
2
10
11
8
7
5
6
9
12
L
Example 1
Naming Pairs of Angles





How is 9 related to the other angles?
9 and 10 are a linear pair. So are 9 and 12
9 and 11 are vertical angles
9 and 7 are alternate exterior angles. So are 9 and 3
9 and 5 are corresponding
angles. So are 9 and 1
n
4
1
m
3
2
10
11
8
7
5
6
9
12
L
PARALLEL LINES AND
CONGRUENT ANGLES
While these definitions apply whenever two lines
are cut by a transversal, we will normally talk
about this for parallel lines
1. What is the Parallel Postulate and how can I
use it?
2.
What are the pairs of congruent angles formed
by parallel lines cut by a transversal?
3.
Lets reexamine the angle definitions for
parallel lines
PARALLEL LINES
Two coplaner lines that do not intersect
The book notes them with “matching
arrows” see page 117
TRANSVERSAL (Parallel)
A line that intersects two coplanar lines in two
distinct points.
SPECIAL ANGLES (parallel)
1
2
4
Interior Angles – lie between
6
the two lines (3, 4, 5, and
5
6)
7 8
Alternate Interior Angles – are
on opposite sides of the
transversal. (3 & 6 AND 4
and 5)
Same-Side Interior Angles – are on the same
side of the transversal. (3 & 5 AND 4 & 6)
3
MORE SPECIAL ANGLES
1 2
34
Exterior Angles – lie
outside the two lines (1, 2,
7, and 8)
5 68
7
Alternate Exterior Angles – are on
opposite sides of the transversal (1& 8
AND 2 & 7)
Corresponding Angles – same location,
different intersections (2 & 6, 4 & 8, 1 &
5, 3 & 7)
PARALLEL LINES AND A TRANSVERSAL
1. On your paper, construct two parallel lines, then
construct an “angled” transversal.
2. Label each angle made ( 1 - 8)
3. Based on appearance, make a conjecture as to
which angles are congruent, supplementary, or
complementary to each other.
4. Using the protractor, measure and label each
angle.
5. Make 4 conjectures about the angle pairs:
( alternate int, consecutive ext, corresponding…)
IF 2 PARALLEL LINES ARE CUT BY A
TRANSVERSAL, THEN ANY PAIR OF THE ANGLES
FORMED ARE EITHER CONGRUENT OR
SUPPLEMENTARY.
Vertical Angles & Linear Pair
Vertical Angles:
Linear Pair:
Two angles that are opposite angles. Vertical
angles are congruent.
 1   4,  2   3,  5   8,  6   7
Supplementary angles that form a line (sum = 180)
1 & 2 , 2 & 4 , 4 &3, 3 & 1,
5 & 6, 6 & 8, 8 & 7, 7 & 5
1
3 4
5
7
6
8
2
Angles and Parallel Lines

If two parallel lines are cut by a transversal, then the following pairs of angles
are congruent.
1.
Corresponding angles
2.
Alternate interior angles
3.
Alternate exterior angles

If two parallel lines are cut by a transversal, then the following pairs of angles
are supplementary.
1.
Same-Side interior angles
2.
Same-Side exterior angles
Continued…..
Corresponding Angles & Consecutive Angles
Corresponding Angles: Two angles that occupy corresponding positions.
 2   6,  1   5,  3   7,  4   8
1
3
5 6
7 8
2
4
Consecutive Angles
Consecutive Interior Angles: Two angles that lie between parallel lines on the same sides
of the transversal.
m3 +m5 = 180º, m4 +m6 = 180º
Consecutive Exterior Angles: Two angles that lie outside parallel lines on the same sides
of the transversal.
1
m1 +m7 = 180º, m2 +m8 = 180º
3
4
5
7
2
6
8
Alternate Angles

Alternate Interior Angles: Two angles that lie between parallel lines on opposite sides
of the transversal (but not a linear pair).
 3   6,  4   5

Alternate Exterior Angles: Two angles that lie outside parallel lines on opposite sides
of the transversal.
 2   7,  1   8
1
3
4
5
7
2
6
8
Example: If line AB is parallel to line CD and s is parallel to t, find the measure
of all the angles when m< 1 = 100°. Justify your answers.
A
1
4
C
5
8
s
2
9
12
3
6
13
16
7
t
10
11
14
15
B
D
Example:
If line AB is parallel to line CD and s is parallel to t, find:
1. the value of x, if m<3 = 4x + 6 and the m<11 = 126.
2. the value of x, if m<1 = 100 and m<8 = 2x + 10.
1. 30
2. 35
3. the value of y, if m<11 = 3y – 5 and m<16 = 2y + 20. 3. 33
A
1
4
C
5
8
s
2
9
12
3
6
13
16
7
t
B
11
14
15
10
D
IF TWO LINES ARE PARALLEL:
1. A pair of corresponding s are .
2. A pair of alternate interior s are .
3. A pair of alternate exterior s are .
4. A pair of consecutive interior s are
supplementary.
5. A pair of consecutive exterior s are
supplementary.
INVESTIGATING POSTULATES
1. Construct a line named AB.
2. Somewhere above or below the line, put a pt P.
3. Construct 2 lines that go through pt P that are
also || to AB
Parallel Postulates

The following have been proven true and can be
applied to parallel lines.
Parallel Postulates
THROUGH A POINT NOT ON A LINE THERE IS
EXACTLY ONE PARALLEL TO THE GIVEN LINE.
INVESTIGATING POSTULATES
1. Construct 2 parallel lines AB and CD.
2. Construct LP so that it is  to AB and also
passes through CD.
3. Measure the angles of the intersection on LP and
CD.
4. What can you conclude?
IN A PLANE, IF A LINE IS PERPENDICULAR TO 1
OF 2 PARALLEL LINES, THEN IT IS
PERPENDICULAR TO THE OTHER.
IF 2 LINES ARE PARALLEL TO A 3RD LINE, THEN
THEY ARE PARALLEL TO EACH OTHER.
(TRANSITIVE PROP. OF || LINES)
Classwork - Go!
118
5-8
10-17
23-25
30
Starting on Pg